But let us still point out the following general property however, valid for any Bateman solution such that \(\sigma ^\mu =0\) 
in (REF ). This property extends to the very specific case of ModMax theories a same property valid for the ordinary Bateman solutions
to the linear Maxwell equations in the form recalled above[1]}, [2]}, [3]}. Namely that given any pair \((\alpha (x^\mu ),\beta (x^\mu ))\) 
of Bateman potentials obeying (REF ) but specifically with \(\sigma ^\mu (x^\mu )=0\) , then any bi-holomorphic complex
transformation \((f(\alpha ,\beta ),g(\alpha ,\beta ))\)  defines
again a pair of Bateman potentials solving once again the ModMax equations in (REF ) and once again with
\(\sigma ^\mu =0\) . This simple observation provides a powerful tool to generate whole classes of new solutions to the ModMax equations starting
from known ones, provided of course that no term in \(\sigma ^\mu \)  be present for the initial solution. The basic simple double reason for this fact is,
on the one hand, the following property,
\(\partial _\mu f\,\partial _\nu g - \partial _\nu \, f \partial _\mu g=\left(\partial _\alpha f \partial _\beta g - \partial _\beta f \partial _\alpha g\right)\left(\partial _\mu \alpha \, \partial _\nu \beta - \partial _\nu \alpha \, \partial _\mu \beta \right),\) 
